Inverse square law of gravitation in (2 + 1)-dimensional space–time as a consequence of Casimir energy

The gravitational effect of vacuum polarization in space exterior to a particle in (2 + 1)-dimensional Einstein theory is investigated. In the weak field limit this gravitational field corresponds to an inverse square law of gravitational attraction, even though the gravitational mass of the quantum vacuum is negative. The paradox is resolved by considering a particle of finite extension and taking into account the vacuum polarization in its interior.